21-259 Calculus in Three Dimensions
Calculus in Three Dimensions is the third course in the Calculus sequence. In the first course, we learn about the derivative and its applications. In the second course, we learn integration techniques, and we make full use of the Fundamental Theorem of Calculus. In this course we will learn about differentiation and integration of functions of more than one variable, and tie these topics together with the study of vector calculus.
First we will spend some time with the basic elements that we'll need to understand: the geometry of three dimensions, lines and planes, functions of 2 (or more) variables, limits and continuity for these functions, and so on. Partial derivatives, which allows us to determine rates of change of a function with respect to each independent variable, will then be introduced. Other differentiation concepts will be studied as well, including the total derivative of a function, the gradient of a function, and the divergence and curl of a field. Various applications of derivatives will again be considered, and most importantly the optimization problems.
Integration will be introduced, and the various types of integrals will depend on the function and domain considered. Keep in mind that the integral is a tool, an extremely flexible and powerful tool to measure and analyze functions. We'll encounter double integrals, triple integrals, line integrals along curves, and surface integrals. Also, we'll integrate vector fields over both oriented curves and surfaces. Fortunately, we'll be able to reduce the calculation of these integrals to one-variable integrals.
Finally, the course reaches its beautiful and natural conclusion with the higher dimensional versions of the Fundamental Theorem of Calculus.
After completing this course, you will have
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